ENEE 241 02*
HOMEWORK ASSIGNMENT 4
Due Fri 02/08
Consider the discretetime sinusoids
x
[
n
] = cos
3
πn
4
+
2
π
3
¶
and
y
[
n
] = cos
πn
5
+
2
π
3
¶
(i) (4 pts.)
What is the period
N
of
x
[
·
]? Use MATLAB to generate a discrete plot of the first
four periods of
x
[
·
], (i.e., for
n
= 0
, . . . ,
4
N

1.)
(ii) (4 pts.)
How many different sinusoids of the form
cos(
ωn
)
are periodic with the same fundamental period as
x
[
·
]? Determine the values of
ω
in [0
,
2
π
) that
produce these sinusoids.
(iii) (4 pts.)
The sequence
v
[
·
] is formed by taking every second sample in
y
[
·
], i.e.,
v
[
n
] =
y
[2
n
]
Write an equation for
v
[
n
]. What is the period of
v
[
·
]?
(iv) (4 pts.)
Use reasoning (possibly aided by MATLAB) to determine whether
x
[
n
] +
y
[
n
]
is periodic. If it is, what is its fundamental period?
(v) (4 pts.)
Is the function
z
[
n
] =
x
[
n
]
2
periodic? If it is, what is its fundamental period?
Solved Examples
S
4.1
. Find all frequencies
ω
in [0
, π
] for which the discretetime sinusoid
x
[
n
] = cos
ωn
is periodic with
fundamental
period
N
= 16.
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 Spring '08
 staff
 Cos, Period, Full stop

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